A053993 The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.

1, 1, 3, 5, 9, 14, 24, 35, 55, 81, 120, 171, 248, 345, 486, 669, 920, 1246, 1690, 2256, 3014, 3984, 5253, 6870, 8970, 11618, 15022, 19306, 24745, 31557, 40154, 50845, 64244, 80850, 101501, 126982, 158514, 197218, 244865, 303143, 374497, 461435

Offset: 0

Author

James Sellers, Apr 04 2000

Keywords

• easy
• nonn

Comments

Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - Vladeta Jovovic, Feb 16 2005

The sequence A077285 is the same but with multiplicities of all parts.

Examples

			1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - _Michael Somos_, Mar 09 2011
		

References

• George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See formula (18) on page 3944.

Crossrefs

Cf. A077285, A052992, A097242, A247661, A247662.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
      +add(b(n-i*j, i-1)*`if`(irem(i, 2)=1, j, 1), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 16 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] * If[Mod[i, 2] == 1, j, 1], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
QP = QPochhammer; s = QP[q^4] * (QP[q^6]^2 / (QP[q] * QP[q^2] * QP[q^3] * QP[q^12])) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* Michael Somos, Mar 09 2011 */

Formula

Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - Michael Somos, Mar 09 2011

Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - Michael Somos, Mar 09 2011

G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*sqrt(2)*n). - Vaclav Kotesovec, Nov 28 2015